In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coefficient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fraction-free algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to compute a greatest common right divisor and a least common left multiple of such matrices. Our fraction-free row reduction algorithm can be viewed as a generalization of subresultant algorithms. The linear algebra formulation allows us to obtain bounds on the size of the intermediate results and to analyze the complexity of our algorithms. We then make use of the fraction-free algorithm as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov form, and the Popov form of a polynomial matrix. By examining the linear algebra formulation, we develop criteria for detecting unlucky homomorphisms and determining the number of homomorphic images required.

In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coefficient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fraction-free algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to compute a greatest common right divisor and a least common left multiple of such matrices. Our fraction-free row reduction algorithm can be viewed as a generalization of subresultant algorithms. The linear algebra formulation allows us to obtain bounds on the size of the intermediate results and to analyze the complexity of our algorithms. We then make use of the fraction-free algorithm as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov form, and the Popov form of a polynomial matrix. By examining the linear algebra formulation, we develop criteria for detecting unlucky homomorphisms and determining the number of homomorphic images required.

The interest in wireless communications among consumers has exploded since the introduction of the "3G" cell phone standards. One reason for their success is the increasingly higher data rates achievable through the networks. A further increase in data rates is possible through the use of multiple antennas at either or both sides of the wireless links. Precoding and receive filtering using matrices obtained from a singular value decomposition (SVD) of the channel matrix is a transmission strategy for achieving the channel capacity of a deterministic narrowband multiple-input multiple-output (MIMO) communications channel. When signalling over wideband channels using orthogonal frequency-division multiplexing (OFDM), an SVD must be performed for every sub-carrier. As the number of sub-carriers of this traditional approach grow large, so does the computational load. It is therefore interesting to study alternate means for obtaining the decomposition. A wideband MIMO channel can be modeled as a matrix filter with a finite impulse response, represented by a polynomial matrix. This thesis is concerned with investigating algorithms which decompose the polynomial channel matrix directly. The resulting decomposition factors can then be used to obtain the sub-carrier based precoding and receive filtering matrices. Existing approximative polynomial matrix QR and singular value decomposition algorithms were modified, and studied in terms of decomposition quality and computational complexity. The decomposition algorithms were shown to give decompositions of good quality, but if the goal is to obtain precoding and receive filtering matrices, the computational load is prohibitive for channels with long impulse responses. Two algorithms for performing exact rational decompositions (QRD/SVD) of polynomial matrices were proposed and analyzed. Although they for simple cases resulted in excellent decompositions, issues with numerical stability of a spectral factorization step renders the algorithms in their current form purposeless. For a MIMO channel with exponentially decaying power-delay profile, the sum rates achieved by employing the filters given from the approximative polynomial SVD algorithm were compared to the channel capacity. It was shown that if the symbol streams were decoded independently, as done in the traditional approach, the sum rates were sensitive to errors in the decomposition. A receiver with a spatially joint detector achieved sum rates close to the channel capacity, but with such a receiver the low complexity detector set-up of the traditional approach is lost. Summarizing, this thesis has shown that a wideband MIMO channel can be diagonalized in space and frequency using OFDM in conjunction with an approximative polynomial SVD algorithm. In order to reach sum rates close to the capacity of a simple channel, the computational load becomes restraining compared to the traditional approach, for channels with long impulse responses.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.

In this thesis, we present efficient deterministic algorithms for polynomial matrix computation problems, including the computation of order basis, minimal kernel basis, matrix inverse, column basis, unimodular completion, determinant, Hermite normal form, rank and rank profile for matrices of univariate polynomials over a field. The algorithm for kernel basis computation also immediately provides an efficient deterministic algorithm for solving linear systems. The algorithm for column basis also gives efficient deterministic algorithms for computing matrix GCDs, column reduced forms, and Popov normal forms for matrices of any dimension and any rank. We reduce all these problems to polynomial matrix multiplications. The computational costs of our algorithms are then similar to the costs of multiplying matrices, whose dimensions match the input matrix dimensions in the original problems, and whose degrees equal the average column degrees of the original input matrices in most cases. The use of the average column degrees instead of the commonly used matrix degrees, or equivalently the maximum column degrees, makes our computational costs more precise and tighter. In addition, the shifted minimal bases computed by our algorithms are more general than the standard minimal bases.

Dans cette thèse, nous étudions des algorithmes pour un problème de recherche de relations à une ou plusieurs variables. Il généralise celui de calculer une solution à un système d’équations linéaires modulaires sur un anneau de polynômes, et inclut par exemple le calcul d’approximants de Hermite-Padé ou d’interpolants bivariés. Plutôt qu’une seule solution, nous nous attacherons à calculer un ensemble de générateurs possédant de bonnes propriétés. Précisément, l’entrée de notre problème consiste en un module de dimension finie spécifié par l’action des variables sur ses éléments, et en un certain nombre d’éléments de ce module ; il s’agit de calculer une base de Gröbner du modules des relations entre ces éléments. En termes d’algèbre linéaire, l’entrée décrit une matrice avec une structure de type Krylov, et il s’agit de calculer sous forme compacte une base du noyau de cette matrice. Nous proposons plusieurs algorithmes en fonction de la forme des matrices de multiplication qui représentent l’action des variables. Dans le cas d’une matrice de Jordan,nous accélérons le calcul d’interpolants multivariés sous certaines contraintes de degré ; nos résultats pour une forme de Frobenius permettent d’accélérer le calcul de formes normales de matrices polynomiales univariées. Enfin, dans le cas de plusieurs matrices denses, nous accélérons le changement d’ordre pour des bases de Gröbner d’idéaux multivariés zéro-dimensionnels. / In this thesis, we study algorithms for a problem of finding relations in one or several variables. It generalizes that of computing a solution to a system of linear modular equations over a polynomial ring, including in particular the computation of Hermite- Padéapproximants and bivariate interpolants. Rather than a single solution, we aim at computing generators of the solution set which have good properties. Precisely, the input of our problem consists of a finite-dimensional module given by the action of the variables on its elements, and of some elements of this module; the goal is to compute a Gröbner basis of the module of syzygies between these elements. In terms of linear algebra, the input describes a matrix with a type of Krylov structure, and the goal is to compute a compact representation of a basis of the nullspace of this matrix. We propose several algorithms in accordance with the structure of the multiplication matrices which specify the action of the variables. In the case of a Jordan matrix, we accelerate the computation of multivariate interpolants under degree constraints; our result for a Frobenius matrix leads to a faster algorithm for computing normal forms of univariate polynomial matrices. In the case of several dense matrices, we accelerate the change of monomial order for Gröbner bases of multivariate zero-dimensional ideals.